First thing I would suggest is aking his doctors if they have any recommendations of specific doctors, clinics or hospitals in the area. Doctors often know or know of those things in other areas because they trained with people who went there, meeting other doctors at conferences, etc.
Second suggestion would be to get a short summary of care from the doctors to take along with you. It should include contact information for his current doctor/clinic/hospital. That way, if anything happens, you will have a quick reference for whoever sees the child in Orlando.
The nearest hospital emergency room is in Celebration and that is where they send people who are in life threatening emergencies because it is closest.
The front desk or First Aid in any park can help you with where to go and how to get there - from Urgent Care/clinics/hospitals and driving directions/calling a cab/ambulance.
If he has an issue in a park, go to First Aid. They are staffed with RNs and will call EMTs if you need more care than they can give.
If you are in a park, but not near First Aid and can't get there, ask any CM to call for help for you.
Keep reminding yourself that emergencies happen every day at WDW parks and resorts. Most people are not even aware of them because WDW does a good job of handling them quickly and quietly.
Last thing I can think of - be prepared in case bleeding starts somewhere else. On our last trip, as we were waiting at the airport gate to come home, we saw a family with a young man who was having a very bad nose bleed. They had nothing except a few tissues with. People were bringing them paper towels from the bathroom and a housekeeping person had cleaning cloths, but that was it until EMTs arrived. We had a package of hand wipes that we gave them because the young man was freaking out about all the dried blood on his hands.
I would suggest having a bag with some supplies, including lots of tissues, some wipes, gloves and a small trash bag.
